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Type-IV 't Hooft Anomalies on the Lattice: Emergent Higher-Categorical Symmetries and Applications to LSM Systems

Tsubasa Oishi, Hiromi Ebisu

Abstract

't Hooft anomalies impose fundamental constraints on quantum matter and often lead to emergent symmetry structures upon gauging. We analyze a lattice model with four global symmetries realizing a mixed anomaly described by $\sim a_1\wedge a_2\wedge a_3\wedge a_4$, where the $a_i$ denote background gauge fields for the global symmetries. Through explicit lattice gauging, we demonstrate the emergence of higher symmetry structures, including 2-group, non-invertible, and higher fusion categorical symmetries. We also provide a field-theoretical understanding of these results. Applying this framework to systems with Lieb-Schultz-Mattis anomalies, obtained by promoting part of the internal symmetries to translational symmetries, we demonstrate that modulated (dipole) symmetries arise as direct counterparts of those in systems with purely internal typeIV anomalies. Importantly, we uncover a qualitatively new feature absent in previously studied modulated symmetries: their realization can become intrinsically defect-dependent. In particular, the emergent symmetry structure changes depending on whether symmetry defects are present. This work establishes a concrete lattice realization of mixed anomalies and reveals a rich structure of emergent symmetries, thereby clarifying their role in constraining quantum phases of matter.

Type-IV 't Hooft Anomalies on the Lattice: Emergent Higher-Categorical Symmetries and Applications to LSM Systems

Abstract

't Hooft anomalies impose fundamental constraints on quantum matter and often lead to emergent symmetry structures upon gauging. We analyze a lattice model with four global symmetries realizing a mixed anomaly described by , where the denote background gauge fields for the global symmetries. Through explicit lattice gauging, we demonstrate the emergence of higher symmetry structures, including 2-group, non-invertible, and higher fusion categorical symmetries. We also provide a field-theoretical understanding of these results. Applying this framework to systems with Lieb-Schultz-Mattis anomalies, obtained by promoting part of the internal symmetries to translational symmetries, we demonstrate that modulated (dipole) symmetries arise as direct counterparts of those in systems with purely internal typeIV anomalies. Importantly, we uncover a qualitatively new feature absent in previously studied modulated symmetries: their realization can become intrinsically defect-dependent. In particular, the emergent symmetry structure changes depending on whether symmetry defects are present. This work establishes a concrete lattice realization of mixed anomalies and reveals a rich structure of emergent symmetries, thereby clarifying their role in constraining quantum phases of matter.

Paper Structure

This paper contains 21 sections, 126 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Emergent symmetries via gauging subgroups
  3. Gauging $\mathbb{Z}_2^A$ : 2-group symmetry
  4. Gauging $\mathbb{Z}_2^A \times \mathbb{Z}_2^B$ : Non-invertible symmetry
  5. Gauging $\mathbb{Z}_2^A \times \mathbb{Z}_2^B \times \mathbb{Z}_2^C$ : 2-Representation category
  6. Field-theoretical interpretation
  7. Gauging $\mathbb{Z}_2^A$ : 2-group symmetry
  8. Gauging $\mathbb{Z}_2^A \times \mathbb{Z}_2^B$ : Non-invertible symmetry
  9. Gauging $\mathbb{Z}_2^A \times \mathbb{Z}_2^B\times\mathbb{Z}_2^C$ : 2-Representation category
  10. LSM anomaly and modulated symmetry from type IV anomaly
  11. Two internal and two translational symmetries
  12. Three internal and one translational symmetries
  13. Conclusion
  14. Topological manipulation on the lattice
  15. Gauging $1$-form symmetry
  16. ...and 6 more sections

Figures (13)

  • Figure 1: (a) The product of CZ terms in the Hamiltonian \ref{['model with type4']}. (b) Configuration of the Gauss law $G_{A,a}$, which is described in \ref{['type4:gauss1']}. (c) Flux term $B_A$ given in \ref{['type4:flatness01']}.
  • Figure 2: (a) A configuration with a $\mathbb{Z}_2^B$ defect (blue line) inserted (b) Action of $U_{D,1}$ around the $\mathbb{Z}_2^B$ defect (c) The projective algebra between $U_C$ and $U_D$ under the defect insertion, which gives rise to a $1$-form symmetry (red line) along the defect.
  • Figure 3: Two triangles close to the defect (blue line connecting $a$ and $c$).
  • Figure 4: (a) Gauss law $G_{B,b}$ given in \ref{['type4:gauss2']}. (b) Flux term $B_B$ defined in \ref{['type4:flatness2']}. (c) The product of CZ terms in the Hamiltonian \ref{['gauged model2 with type4']}.
  • Figure 5: (a) A configuration with a $\mathbb{Z}_2^C$ defect (blue line) inserted (b)Action of $U_{D,2}$ around the $\mathbb{Z}_2^C$ defect
  • ...and 8 more figures